Question

A group of students met in the lobby of their school. First, 1/3 of the students were sent to the gym. Then, 1/3 of the remaining students were sent to the cafeteria. Finally, 1/3 of the students still in the lobby were sent to the auditorium, and all the remaining students were sent home. Which of the following could not have been the number of students who originally met in the school lobby?

81
135
162
207
243

Answer 1

let no. of students=x
students sent to the gym=x/3
students remaining=x-x/3=2x/3
students sent to cafteria\[=\frac{ 1 }{ 3 }\times \frac{ 2x }{ 3 }=\frac{ 2x }{ 9 }\]
remaining students\[=\frac{ 2x }{ 3 }-\frac{ 2x }{ 9 }=\frac{ 4x }{ 9 }\]
students sent to auditorium\[=\frac{ 1 }{ 3 }\times \frac{ 4x }{ 9 }=\frac{ 4x }{ 27 }\]
students sent home\[=\frac{ 4x }{ 9 }-\frac{ 4x }{ 27 }=\frac{ 8x }{ 27 }\]
as no. of students is a natural no. so x should be a multiple of 27
81=27*3
135=27*5
162=27*6
207= 23*9
243=27*9
so 207 is not a multiple of 27
hence 207 is the reqd. no. of students